# 1.2 2d dft

 Page 1 / 1
This module extends the ideas of the Discrete Fourier Transform (DFT) into two-dimensions, which is necessary for any image processing.

## 2d dft

To perform image restoration (and many other useful image processing algorithms) in a computer, we need a FourierTransform (FT) that is discrete and two-dimensional.

$F(k, l)=(u=\frac{2\pi k}{N}, v=\frac{2\pi l}{N}, F(u, v))$
for $k=\{0, , N-1\}$ and $l=\{0, , N-1\}$ .
$F(u, v)=\sum \sum f(m, n)e^{-(ium)}e^{-(ivm)}$
$F(k, l)=\sum_{m=0}^{N-1} \sum_{n=0}^{N-1} f(m, n)e^{-i\frac{2\pi km}{N}}e^{-i\frac{2\pi ln}{N}}$
where the above equation ( ) has finite support for an $N$ x $N$ image.

## Inverse 2d dft

As with our regular fourier transforms, the 2D DFT also has an inverse transform that allows us to reconstruct an imageas a weighted combination of complex sinusoidal basis functions.

$f(m, n)=\frac{1}{N^{2}}\sum_{k=0}^{N-1} \sum_{l=0}^{N-1} F(k, l)e^{\frac{i\times 2\pi km}{N}}e^{\frac{i\times 2\pi ln}{N}}$

## 2d dft and convolution

The regular 2D convolution equation is

$g(m, n)=\sum_{k=0}^{N-1} \sum_{l=0}^{N-1} f(k, l)h(m-k, n-l)$

Below we go through the steps of convolving two two-dimensional arrays. You can think of $f$ as representing an image and $h$ represents a PSF, where $h(m, n)=0$ for $m\land n> 1$ and $m\land n< 0$ . $h=\begin{pmatrix}h(0, 0) & h(0, 1)\\ h(1, 0) & h(1, 1)\\ \end{pmatrix}$ $f=\begin{pmatrix}f(0, 0) & & f(0, N-1)\\ & & \\ f(N-1, 0) & & f(N-1, N-1)\\ \end{pmatrix}$ Step 1 (Flip $h$ ):

$h(-m, -n)=\begin{pmatrix}h(1, 1) & h(1, 0) & 0\\ h(0, 1) & h(0, 0) & 0\\ 0 & 0 & 0\\ \end{pmatrix}$
Step 2 (Convolve):
$g(0, 0)=h(0, 0)f(0, 0)$
We use the standard 2D convolution equation ( ) to find the first element of our convolved image. In order to better understand what ishappening, we can think of this visually. The basic idea is to take $h(-m, -n)$ and place it "on top" of $f(k, l)$ , so that just the bottom-right element, $h(0, 0)$ of $h(-m, -n)$ overlaps with the top-left element, $f(0, 0)$ , of $f(k, l)$ . Then, to get the next element of our convolved image, we slide the flipped matrix, $h(-m, -n)$ , over one element to the right and get the following result: $g(0, 1)=h(0, 0)f(0, 1)+h(0, 1)f(0, 0)$ We continue in this fashion to find all of the elements ofour convolved image, $g(m, n)$ . Using the above method we define the general formula to find a particular element of $g(m, n)$ as:
$g(m, n)=h(0, 0)f(m, n)+h(0, 1)f(m, n-1)+h(1, 0)f(m-1, n)+h(1, 1)f(m-1, n-1)$

## Circular convolution

What does $H(k, l)F(k, l)$ produce?

2D Circular Convolution

$\stackrel{~}{g}(m, n)=\mathrm{IDFT}(H(k, l)F(k, l))=\mathrm{circularconvolutionin2D}$

Due to periodic extension by DFT ( ):

Based on the above solution, we will let

$\stackrel{~}{g}(m, n)=\mathrm{IDFT}(H(k, l)F(k, l))$
Using this equation, we can calculate the value for each position on our final image, $\stackrel{~}{g}(m, n)$ . For example, due to the periodic extension of the images, when circular convolution is applied we willobserve a wrap-around effect.
$\stackrel{~}{g}(0, 0)=h(0, 0)f(0, 0)+h(1, 0)f(N-1, 0)+h(0, 1)f(0, N-1)+h(1, 1)f(N-1, N-1)$
Where the last three terms in are a result of the wrap-around effect caused by the presence of the images copies located all around it.

If the support of $h$ is $M$ x $M$ and $f$ is $N$ x $N$ , then we zero pad $f$ and $h$ to $M+N-1$ x $M+N-1$ (see ).

Circular Convolution = Regular Convolution

## Computing the 2d dft

$F(k, l)=\sum_{m=0}^{N-1} \sum_{n=0}^{N-1} f(m, n)e^{-i\frac{2\pi km}{N}}e^{-i\frac{2\pi ln}{N}}$
where in the above equation, $\sum_{n=0}^{N-1} f(m, n)e^{-i\frac{2\pi ln}{N}}$ is simply a 1D DFT over $n$ . This means that we will take the 1D FFT of each row; if wehave $N$ rows, then it will require $N\lg N$ operations per row. We can rewrite this as
$F(k, l)=\sum_{m=0}^{N-1} {f}^{}(m, l)e^{-i\frac{2\pi km}{N}}$
where now we take the 1D FFT of each column, which means that if we have $N$ columns, then it requires $N\lg N$ operations per column.
Therefore the overall complexity of a 2D FFT is $O(N^{2}\lg N)$ where $N^{2}$ equals the number of pixels in the image.

How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
How can I make nanorobot?
Lily
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
how can I make nanorobot?
Lily
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
Got questions? Join the online conversation and get instant answers!